Illustration of Mr.ASH and Mr.ASHPen on some special cases
- About
- 1. Semi-dense, high-dimensional setting
- 2. Correlated variables
- 3. Trendfiltering with 20 changepoints
- 4. Trendfiltering with linear basis function
About
In this demo, I illustrate the convergence of three different algorithms for using Mr.ASH in sparse multiple linear regression and trendfiltering. Given response variables $\mathbf{y}$ for $N$ samples and explanatory variables $\mathbf{X}$ for $P$ variables (generally $N \lt P$). We will perform a sparse multiple regression using the adaptive shrinkage prior (Mr.ASH),
$\mathbf{y} = \mathbf{X}\mathbf{b} + \mathbf{e}$,
$\mathbf{e} \sim \mathcal{N}\left(\mathbf{0} \mid \sigma^2 I_n \right)$,
$\mathbf{b} \sim p\left(\mathbf{b} \mid \boldsymbol{\theta}_1\right)$.
$p\left(b_i \mid \boldsymbol{\theta}_1\right) = \sum_{k=1}^{K} w_k \mathcal{N}\left(b_i \mid \mu_k, \sigma_k^2\right)$ with constraints $w_k \ge 0$, $\sum_{k=1}^{K} w_k = 1$
We assume $\sigma_k$ is known and will estimate ($\mathbf{b}, \mathbf{w}, \sigma^2)$ from the data.
Here, I will compare four methods:
-
mr.ash.alpha. Co-ordinate ascent algorithm for maximizing ELBO (as implemented in
mr.ash.alpha; Github) -
mr.ash.pen. Penalized linear regression using gradient descent (L-BFGS-B) algorithm (as implemented in
mr.ash.pen; Github). - mr.ash.pen (EM). Hybrid algorithm which iterates between (i) estimating $\mathbf{b}$ by minimizing PLR objective (approzimate E-Step), and (ii) estimating $\mathbf{w}$ and $\sigma^2$ by maximizing ELBO (approximate M-step).
-
mr.ash.alpha (init). Same as
mr.ash.alpha, but initialized with the results frommr.ash.pen.
I will illustrate four examples, where mr.ash.alpha is known to perform poorly.
Note: I will use the ELBO at each iteration for comparing the methods, although it is to be noted that the objective function for mr.ash.pen and the objective function for the E-step of mr.ash.pen (EM) are different from the ELBO. Hence, the ELBO will not be monotonically increasing for these two methods. Also, the varobj obtained from mr.ash.alpha is an approximation to the true ELBO and matches with the true ELBO close to the optimum.
import numpy as np
import pandas as pd
from mrashpen.inference.penalized_regression import PenalizedRegression as PLR
from mrashpen.inference.mrash_wrapR import MrASHR
from mrashpen.models.plr_ash import PenalizedMrASH
from mrashpen.models.normal_means_ash_scaled import NormalMeansASHScaled
from mrashpen.inference.ebfit import ebfit
from mrashpen.inference import lbfgsfit
from mrashpen.utils import R_lasso
from mrashpen.inference import elbo as elbo_py
from mrashpen.inference import coordinate_descent_step as cd_step
from mrashpen.models import mixture_gaussian as mix_gauss
import sys
sys.path.append('/home/saikat/Documents/work/sparse-regression/simulation/eb-linreg-dsc/dsc/functions')
import simulate
import matplotlib.pyplot as plt
from pymir import mpl_stylesheet
from pymir import mpl_utils
mpl_stylesheet.banskt_presentation(splinecolor = 'black')
def center_and_scale(Z):
dim = Z.ndim
if dim == 1:
Znew = Z / np.std(Z)
Znew = Znew - np.mean(Znew)
elif dim == 2:
Znew = Z / np.std(Z, axis = 0)
Znew = Znew - np.mean(Znew, axis = 0).reshape(1, -1)
return Znew
def initialize_ash_prior(k, scale = 2, sparsity = None):
w = np.zeros(k)
w[0] = 1 / k if sparsity is None else sparsity
w[1:(k-1)] = np.repeat((1 - w[0])/(k-1), (k - 2))
w[k-1] = 1 - np.sum(w)
sk2 = np.square((np.power(scale, np.arange(k) / k) - 1))
prior_grid = np.sqrt(sk2)
return w, prior_grid
def plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue, bhat,
intercept = 0, title = None):
ypred = np.dot(Xtest, bhat) + intercept
fig = plt.figure(figsize = (12, 6))
ax1 = fig.add_subplot(121)
ax2 = fig.add_subplot(122)
ax1.scatter(ytest, ypred, s = 2, alpha = 0.5)
mpl_utils.plot_diag(ax1)
ax2.scatter(btrue, bhat)
mpl_utils.plot_diag(ax2)
ax1.set_xlabel("Y_test")
ax1.set_ylabel("Y_predicted")
ax2.set_xlabel("True b")
ax2.set_ylabel("Predicted b")
if title is not None:
fig.suptitle(title)
plt.tight_layout()
plt.show()
def plot_convergence(objs, methods, nwarm, eps = 1e-8):
fig = plt.figure(figsize = (12, 6))
ax1 = fig.add_subplot(111)
objmin = np.min([np.min(x) for x in objs])
for obj, method, iteq in zip(objs, methods, nwarm):
m_obj = obj[iteq:] - objmin
m_obj = m_obj[m_obj > 0]
ax1.plot(range(iteq, len(m_obj) + iteq), np.log10(m_obj), label = method)
ax1.legend()
ax1.set_xlabel("Number of Iterations")
ax1.set_ylabel("log( max(ELBO) - ELBO )")
plt.show()
return
def plot_trendfilter_mrashpen(X, y, beta, ytest, bhat,
intercept = 0, title = None):
n = y.shape[0]
p = X.shape[1]
ypred = np.dot(X, bhat) + intercept
fig = plt.figure(figsize = (12, 6))
ax1 = fig.add_subplot(121)
ax2 = fig.add_subplot(122)
ax1.scatter(np.arange(n), ytest, edgecolor = 'black', facecolor='white')
ax1.plot(np.arange(n), ypred)
ax1.set_xlabel("Sample index")
ax1.set_ylabel("Y")
ax2.scatter(np.arange(p), beta, edgecolor = 'black', facecolor = 'white')
ax2.scatter(np.arange(p), bhat, s = 40, color = 'firebrick')
ax2.set_xlabel("Sample index")
ax2.set_ylabel("b")
if title is not None:
fig.suptitle(title)
plt.tight_layout()
plt.show()
def linreg_summary_df(sigma2, objs, methods):
data = [[strue * strue, '-', '-']]
rownames = ['True']
for obj, method in zip(objs, methods):
data.append([obj.residual_var, obj.elbo_path[-1], obj.niter])
rownames.append(method)
colnames = ['sigma2', 'ELBO', 'niter']
df = pd.DataFrame.from_records(data, columns = colnames, index = rownames)
return df
1. Semi-dense, high-dimensional setting
In high-dimensional data ($N = 200$ and $P = 2000$), Mr.ASH is known perform worse than LASSO in a dense setting with large number of non-causal variables and high PVE (1, 2). Here, I used $P_{\textrm{causal}} = 500$ and PVE $= 0.95$. Here, mr.ash.pen and mr.ash.pen (EM) converges to a better solution, the first one with a significantly higher ELBO and the second one with a slightly higher ELBO (!).
Note: I found that mr.ash.pen (EM) converges to the correct solution for a wider range of $P_{\textrm{causal}}$. (To do: Run simulation with DSC to explore the full range)
n = 200
p = 2000
p_causal = 50
pve = 0.95
k = 20
X, y, Xtest, ytest, btrue, strue = simulate.equicorr_predictors(n, p, p_causal, pve, rho = 0.0, seed = 100)
X = center_and_scale(X)
Xtest = center_and_scale(Xtest)
wk, sk = initialize_ash_prior(k, scale = 2)
Here, I run the three methods.
'''
Lasso initialization
'''
lasso_a0, lasso_b, _ = R_lasso.fit(X, y)
s2init = np.var(y - np.dot(X, lasso_b) - lasso_a0)
winit = mix_gauss.emfit(lasso_b, sk)
print ("Lasso initialization")
'''
mr.ash.pen
'''
plr_lbfgs = PLR(method = 'L-BFGS-B', is_prior_scaled = True,
debug = False, display_progress = False, calculate_elbo = True,
maxiter = 2000, tol = 1e-8)
plr_lbfgs.fit(X, y, sk, binit = lasso_b, winit = winit, s2init = s2init)
# plr_lbfgs.fit(X, y, sk, binit = None, winit = wk, s2init = s2init)
# plr_lbfgs = lbfgsfit(X, y, sk, wk, binit = lasso_b, s2init = s2init, calculate_elbo = True)
'''
mr.ash.pen (EM)
'''
# plr_eb = ebfit(X, y, sk, wk, binit = None, s2init = 1, maxiter = 200, qb_maxiter = 100)
plr_eb = ebfit(X, y, sk, binit = lasso_b, winit = winit, s2init = s2init,
maxiter = 200, qb_maxiter = 50, calculate_elbo = True)
'''
mr.ash.alpha
'''
mrash_r = MrASHR(option = "r2py", debug = False)
mrash_r.fit(X, y, sk, binit = lasso_b, winit = winit, s2init = s2init)
'''
mr.ash.alpha (init)
'''
mrash_r_init = MrASHR(option = "r2py", debug = False)
mrash_r_init.fit(X, y, sk, binit = plr_lbfgs.coef, winit = plr_lbfgs.prior, s2init = plr_lbfgs.residual_var)
On the left panel I compare the predicted $\mathbf{y}_{\mathrm{test}}$ by the different methods (y-axis) with the true $\mathbf{y}_{\mathrm{test}}$ (x-axis). On the right panel, I compare the predicted coefficients (y-axis) with their true values (x-axis). The plot at the bottom shows the convergence of the different methods against the number of iteration.
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
lasso_b, intercept = lasso_a0, title = 'Lasso')
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
mrash_r.coef, intercept = mrash_r.intercept, title = 'mr.ash.alpha')
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
plr_lbfgs.coef, intercept = plr_lbfgs.intercept, title = 'mr.ash.pen')
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
plr_eb.coef, intercept = plr_eb.intercept, title = 'mr.ash.pen (EM)')
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
mrash_r_init.coef, intercept = mrash_r_init.intercept, title = 'mr.ash.alpha (init)')
kinit = [0, 0, 0]
objs = [mrash_r.obj_path, plr_lbfgs.elbo_path, plr_eb.elbo_path]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)"]
plot_convergence(objs, methods, kinit)
objs = [mrash_r, plr_lbfgs, plr_eb, mrash_r_init]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)", "mr.ash.alpha (init)"]
df = linreg_summary_df(strue, objs, methods)
df
2. Correlated variables
It was also found earlier that mr.ash.alpha performs suboptimally, when the variable are correlated (1). This is an interesting case, because real world applications on genetic data involves strong correlations. Here, I have generated data from matrix normal distribution, $\mathbf{X} \sim \mathcal{MN}\left(0, \mathbb{I}_N, \mathbf{\Sigma}_{\rho} \right)$, where $\mathbf{\Sigma}_{\rho}$ is a matrix with ones along the diagonal and all off diagonal entries set to $\rho \gt 0$. That is, each row of $\mathbf{X}$ is multivariate normal with zero mean and covariance $\mathbf{\Sigma}_{\rho}$. In this particular example, I have used $\rho = 0.95$ and PVE = $0.95$ with 10 non-causal variables in a high-dimension setting ($N = 200$ and $P = 2000$).
Interestingly, the convergence for mr.ash.pen happens after a large number of iterations, while the other methods fails to converge to a higher ELBO.
n = 200
p = 2000
p_causal = 10
pve = 0.7
rho = 0.95
k = 20
X, y, Xtest, ytest, btrue, strue = simulate.equicorr_predictors(n, p, p_causal, pve, rho = rho, seed = 10)
X = center_and_scale(X)
Xtest = center_and_scale(Xtest)
wk, sk = initialize_ash_prior(k, scale = 2)
'''
Lasso initialization
'''
lasso_a0, lasso_b, _ = R_lasso.fit(X, y)
s2init = np.var(y - np.dot(X, lasso_b) - lasso_a0)
winit = mix_gauss.emfit(lasso_b, sk)
print ("Lasso initialization")
'''
mr.ash.pen
'''
plr_lbfgs = PLR(method = 'L-BFGS-B', is_prior_scaled = True,
debug = False, display_progress = False, calculate_elbo = True,
maxiter = 2000, tol = 1e-8, function_call = 'python')
plr_lbfgs.fit(X, y, sk, binit = lasso_b, winit = winit, s2init = s2init)
#plr_lbfgs = lbfgsfit.method_fixseq_simple(X, y, sk, wk, binit = lasso_b, s2init = s2init, calculate_elbo = True)
#plr_lbfgs = lbfgsfit.method_init_gb(X, y, sk, wk, binit = lasso_b, s2init = s2init, calculate_elbo = True)
'''
mr.ash.pen (EM)
'''
plr_eb = ebfit(X, y, sk, binit = lasso_b, winit = winit, s2init = s2init,
maxiter = 200, qb_maxiter = 50, calculate_elbo = True)
'''
mr.ash.alpha
'''
mrash_r = MrASHR(option = "r2py", debug = False)
mrash_r.fit(X, y, sk, binit = lasso_b, winit = winit, s2init = s2init)
'''
mr.ash.alpha (init)
'''
mrash_r_init = MrASHR(option = "r2py", debug = False)
mrash_r_init.fit(X, y, sk, binit = plr_lbfgs.coef, winit = plr_lbfgs.prior, s2init = plr_lbfgs.residual_var)
'''
Plot
'''
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
lasso_b, intercept = lasso_a0, title = 'Lasso')
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
mrash_r.coef, intercept = mrash_r.intercept, title = 'mr.ash.alpha')
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
plr_lbfgs.coef, intercept = plr_lbfgs.intercept, title = 'mr.ash.pen')
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
plr_eb.coef, intercept = plr_eb.intercept, title = 'mr.ash.pen (EM)')
plot_linear_mrashpen(X, y, Xtest, ytest, btrue, strue,
mrash_r_init.coef, intercept = mrash_r_init.intercept, title = 'mr.ash.alpha (init)')
kinit = [0, 0, 0]
objs = [mrash_r.obj_path, plr_lbfgs.elbo_path, plr_eb.elbo_path]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)"]
plot_convergence(objs, methods, kinit)
objs = [mrash_r, plr_lbfgs, plr_eb, mrash_r_init]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)", "mr.ash.alpha (init)"]
df = linreg_summary_df(strue, objs, methods)
df
3. Trendfiltering with 20 changepoints
Another interesting and relatively harder problem is trendfiltering. Previously, I looked at many different cases of trendfiltering with mr.ash.alpha (link). For the zero-th order trendfiltering ($k = 0$), I found that Mr.ASH performance becomes worse with increasing the number of changepoints, $s$. Here, I show an example with $s = 20$.
n = 200
p = 200
p_causal = 20
snr = 10
k = 20
X, y, Xtest, ytest, btrue, strue = simulate.changepoint_predictors (n, p, p_causal, snr,
k = 0, signal = 'gamma', seed = 100)
wk, sk = initialize_ash_prior(k, scale = 4)
'''
Lasso initialization
'''
genlasso_a0, genlasso_b, _ = R_genlasso.fit(X, y, order = 0)
genlasso_b = genlasso_b.reshape(-1,)
s2init = np.var(y - np.dot(X, genlasso_b) - genlasso_a0)
winit = mix_gauss.emfit(genlasso_b, sk)
print ("Lasso initialization")
'''
mr.ash.pen
'''
plr_lbfgs = PLR(method = 'L-BFGS-B', is_prior_scaled = True,
debug = False, display_progress = False, calculate_elbo = True,
maxiter = 2000, tol = 1e-8)
plr_lbfgs.fit(X, y, sk, binit = genlasso_b, winit = wk, s2init = s2init)
'''
mr.ash.pen (EM)
'''
plr_eb = ebfit(X, y, sk, binit = genlasso_b, winit = wk, s2init = s2init,
maxiter = 200, qb_maxiter = 50, calculate_elbo = True)
'''
mr.ash.alpha
'''
mrash_r = MrASHR(option = "r2py", debug = False)
mrash_r.fit(X, y, sk, binit = genlasso_b, winit = wk, s2init = s2init)
'''
mr.ash.alpha (init)
'''
mrash_r_init = MrASHR(option = "r2py", debug = False)
mrash_r_init.fit(X, y, sk, binit = plr_lbfgs.coef, winit = plr_lbfgs.prior, s2init = plr_lbfgs.residual_var)
On the left panel I compare the predicted $\mathbf{y}_{\mathrm{test}}$ by the different methods (solid blue line) with the true $\mathbf{y}_{\mathrm{test}}$ (empty black circles). On the right panel, I compare the predicted coefficients (solid red circles) with their true values (empty black circles). The plot at the bottom shows the convergence of the different methods against the number of iteration.
'''
Plot
'''
plot_trendfilter_mrashpen(X, y, btrue, ytest,
genlasso_b, intercept = genlasso_a0, title = 'Lasso')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
mrash_r.coef, intercept = mrash_r.intercept, title = 'mr.ash.alpha')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
plr_lbfgs.coef, intercept = plr_lbfgs.intercept, title = 'mr.ash.pen')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
plr_eb.coef, intercept = plr_eb.intercept, title = 'mr.ash.pen (EM)')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
mrash_r_init.coef, intercept = mrash_r_init.intercept, title = 'mr.ash.alpha (init)')
kinit = [0, 0, 0]
objs = [mrash_r.obj_path, plr_lbfgs.elbo_path, plr_eb.elbo_path]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)"]
plot_convergence(objs, methods, kinit)
objs = [mrash_r, plr_lbfgs, plr_eb, mrash_r_init]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)", "mr.ash.alpha (init)"]
df = linreg_summary_df(strue, objs, methods)
df
4. Trendfiltering with linear basis function
With a linear basis function, Mr.ASH tends to perform worse than other sparse regression methods. However, in this case, mr.ash.pen and mr.ash.pen (EM) also performs poorly. I found that the poor performance is mainly due to a low spread of variance in the mixture prior. As illustrated below, the performance of mr.ash.pen improves by changing the mixture prior.
Note that mr.ash.pen does not converge after 1000 iterations.
n = 200
p = 200
p_causal = 4
snr = 200
k = 10
X, y, Xtest, ytest, btrue, strue = simulate.changepoint_predictors (n, p, p_causal, snr,
k = 1, signal = 'gamma', seed = 100)
wk, sk = initialize_ash_prior(k, scale = 2)
'''
Lasso initialization
'''
genlasso_a0, genlasso_b, _ = R_genlasso.fit(X, y, order = 1)
genlasso_b = genlasso_b.reshape(-1,)
s2init = np.var(y - np.dot(X, genlasso_b) - genlasso_a0)
winit = mix_gauss.emfit(genlasso_b, sk)
print ("Lasso initialization")
'''
mr.ash.pen
'''
plr_lbfgs = PLR(method = 'L-BFGS-B', is_prior_scaled = True,
debug = False, display_progress = False, calculate_elbo = True,
maxiter = 2000, tol = 1e-8)
plr_lbfgs.fit(X, y, sk, binit = genlasso_b, winit = wk, s2init = s2init)
'''
mr.ash.pen (EM)
'''
plr_eb = ebfit(X, y, sk, binit = genlasso_b, winit = wk, s2init = s2init,
maxiter = 200, qb_maxiter = 50, calculate_elbo = True)
'''
mr.ash.alpha
'''
mrash_r = MrASHR(option = "r2py", debug = False)
mrash_r.fit(X, y, sk, binit = genlasso_b, winit = wk, s2init = s2init)
'''
mr.ash.alpha (init)
'''
mrash_r_init = MrASHR(option = "r2py", debug = False)
mrash_r_init.fit(X, y, sk, binit = plr_lbfgs.coef, winit = plr_lbfgs.prior, s2init = plr_lbfgs.residual_var)
'''
Plot
'''
plot_trendfilter_mrashpen(X, y, btrue, ytest,
genlasso_b, intercept = genlasso_a0, title = 'Lasso')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
mrash_r.coef, intercept = mrash_r.intercept, title = 'mr.ash.alpha')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
plr_lbfgs.coef, intercept = plr_lbfgs.intercept, title = 'mr.ash.pen')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
plr_eb.coef, intercept = plr_eb.intercept, title = 'mr.ash.pen (EM)')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
mrash_r_init.coef, intercept = mrash_r_init.intercept, title = 'mr.ash.alpha (init)')
kinit = [0, 0, 0]
objs = [mrash_r.obj_path, plr_lbfgs.elbo_path, plr_eb.elbo_path]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)"]
plot_convergence(objs, methods, kinit)
objs = [mrash_r, plr_lbfgs, plr_eb, mrash_r_init]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)", "mr.ash.alpha (init)"]
df = linreg_summary_df(strue, objs, methods)
df
wk, sk = initialize_ash_prior(k, scale = 100)
Here, none of the methods converge after reaching their maximum allowed iterations. However, mr.ash.pen provides better estimate for the coefficients, but surprisingly the ELBO is lower than that of mr.ash.alpha.
'''
mr.ash.pen
'''
plr_lbfgs = PLR(method = 'L-BFGS-B', is_prior_scaled = True,
debug = False, display_progress = False, calculate_elbo = True,
maxiter = 2000, tol = 1e-8)
plr_lbfgs.fit(X, y, sk, binit = genlasso_b, winit = wk, s2init = s2init)
'''
mr.ash.pen (EM)
'''
plr_eb = ebfit(X, y, sk, binit = genlasso_b, winit = wk, s2init = s2init,
maxiter = 200, qb_maxiter = 50, calculate_elbo = True)
'''
mr.ash.alpha
'''
mrash_r = MrASHR(option = "r2py", debug = False)
mrash_r.fit(X, y, sk, binit = genlasso_b, winit = wk, s2init = s2init)
'''
mr.ash.alpha (init)
'''
mrash_r_init = MrASHR(option = "r2py", debug = False)
mrash_r_init.fit(X, y, sk, binit = plr_lbfgs.coef, winit = plr_lbfgs.prior, s2init = plr_lbfgs.residual_var)
'''
Plot
'''
plot_trendfilter_mrashpen(X, y, btrue, ytest,
genlasso_b, intercept = genlasso_a0, title = 'Lasso')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
mrash_r.coef, intercept = mrash_r.intercept, title = 'mr.ash.alpha')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
plr_lbfgs.coef, intercept = plr_lbfgs.intercept, title = 'mr.ash.pen')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
plr_eb.coef, intercept = plr_eb.intercept, title = 'mr.ash.pen (EM)')
plot_trendfilter_mrashpen(X, y, btrue, ytest,
mrash_r_init.coef, intercept = mrash_r_init.intercept, title = 'mr.ash.alpha (init)')
kinit = [0, 0, 0]
objs = [mrash_r.obj_path, plr_lbfgs.elbo_path, plr_eb.elbo_path]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)"]
plot_convergence(objs, methods, kinit)
objs = [mrash_r, plr_lbfgs, plr_eb, mrash_r_init]
methods = ["mr.ash.alpha", "mr.ash.pen", "mr.ash.pen (EM)", "mr.ash.alpha (init)"]
df = linreg_summary_df(strue, objs, methods)
df